Weighing Inquiry-Based Learning and Direct Instruction in Elementary Math

Direct instruction or inquiry-based learning? Direct instruction is carefully sequenced, clear, and explicit. Inquiry-based learning promotes higher-level thinking and deep understanding. Research tells us that both approaches have value, but when should we use one instead of the other? 

One factor to consider is the subject. In math, students need opportunities to work on rich tasks and solve problems in ways that make sense to them. However, that doesn’t mean direct instruction is totally absent from math time. The questions below can guide you in deciding whether to use direct instruction, when it would be appropriate, and who else in the classroom you might involve.

1. Was this knowledge discovered or decided?

In deciding whether to use direct instruction, ask yourself whether the thing you want students to learn today was discovered (a concept) or decided (a convention). If the knowledge was discovered as a concept by mathematicians of the past, then students deserve opportunities to discover it in the present. If the knowledge was decided on as a convention by mathematicians of the past, students need to be told what the decision was.

Classroom example: Rounding. The idea of rounding, or approximating numbers according to a specified place value, was discovered. For centuries and across civilizations, people have found it useful to make numbers less precise. Students may experience this when measuring: While textbook examples are often an exact number of units long, students measuring in a real-life inquiry will likely have to round. 

Although rounding is a discoverable concept, the vocabulary word and how we use it is a convention. I draw an open number line labeled with 40 and 50 and ask students whether 42 should be drawn closer to 40 or 50, leaning on their conceptual knowledge of estimating. Then I can make a connection to the conventional language of rounding: “We call this rounding to the nearest ten.”

But what about 45? Is that closer to 40 or 50? Here, students need to be introduced to another convention: Mathematicians decided that if there’s a “tie,” we round to the bigger number. Students will never discover this rule about rounding on their own; it was decided, and that’s exactly what we can tell them through direct instruction.

2. What return do I expect on this investment?

In deciding when to use direct instruction, ask yourself what kind of return you expect on the investment of explicitly telling this knowledge. How much do you expect the direct instruction you’re planning will be “paid back to you” in the students’ subsequent inquiry-based learning?

The idea of a “knowledge loan” comes from the early childhood education centers of Reggio Emilia, Italy, and schools inspired by their approach. For example, as young children explore using clay, they may come to a point where they want to join pieces together. The teacher can lend them knowledge of how sculptors use slip (watery clay) to join pieces together. This is a small investment with huge returns, as the children go on to join parts together to create many original works of art.

Classroom example: Parentheses. Like the children working with clay, children working with numbers may come to a point where they want to join expressions and operations together. 

When students are trying to explain how they used a series of steps to arrive at their answer, and they can confidently represent each step with separate expressions or equations, I might show them how to use parentheses to join the steps into one equation: “So you’re telling me that you knew 5 x 12 would be the same as 5 x 10, that goes together, plus 5 x 2, and that goes together?” Meanwhile, I would write 5 x 12 = (5 x 10) + (5 x 2). “Mathematicians use these (pointing to the parentheses) to show that this part goes together.”

This is lending them a little bit of knowledge, with big expected returns on their ability to represent their thinking going forward.

3. Are there students I can position as experts?

The teacher’s role in inquiry-based learning is sometimes characterized as “guide on the side,” whereas in direct instruction the teacher is characterized as the “sage on the stage.” If you have already decided that direct instruction is the right approach and that now is the right time, you may also ask who else you might bring on the “stage” with you.

Skilled teachers strategically position students as experts. Especially in math, a major part of our job is ensuring that students maintain their confidence—that they see math as a subject that makes sense and they identify as capable of making sense of it. 

Classroom example: Different interactions. In a whole group setting, there might be a student who already knows or uses the convention I want to directly teach. I can present their work for discussion as the day’s warm-up.

In collaboration, when one group is stuck, another group often has the knowledge they need. When a group of second graders was having trouble coming up with an equation to fit  __ + __ -__ = 10, I showed them how another group tried 8 + 4 - 2 = 10. Then I coached them through the logic of this solution and asked what they might try that would be similar. 

By naming, citing, and centering other students as experts, we can meet dual goals of students receiving direct instruction when they need it and facilitating a community of empowered learners. This includes paying special attention to positioning students who may not by default be regarded as or consider themselves experts.

A spectrum of support

In practice, the choice between direct instruction and inquiry-based learning is not either/or. This goes beyond strategically selecting which mode to use and when: Skilled teachers operate all along a spectrum, including instructional approaches that lie somewhere between direct instruction and inquiry-based learning. For more on this spectrum, see chapter 4 of Rachel Lambert’s Rethinking Disability and Mathematics.

At the end of the day, what students will benefit from most is not simply direct instruction or inquiry-based instruction, but rather our thoughtfulness in moving between approaches, from lesson to lesson and interaction to interaction. This is the art and science of our work.